Question

question
in δtuv, u = 380 inches, m∠v = 151°, and m∠t = 25°. find the length of t, to the nearest tenth of an inch.
answer attempt 2 out of 3
t =
inches
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Explanation:

Step1: Find angle U

In a triangle, the sum of angles is \(180^\circ\). So \(m\angle U = 180^\circ - 151^\circ - 25^\circ = 4^\circ\).

Step2: Apply the Law of Sines

The Law of Sines states \(\frac{t}{\sin T}=\frac{u}{\sin U}\). We know \(u = 380\), \(m\angle T = 25^\circ\), \(m\angle U = 4^\circ\). Plugging in: \(\frac{t}{\sin 25^\circ}=\frac{380}{\sin 4^\circ}\).

Step3: Solve for t

\(t=\frac{380\times\sin 25^\circ}{\sin 4^\circ}\). Calculate \(\sin 25^\circ\approx0.4226\), \(\sin 4^\circ\approx0.0698\). Then \(t=\frac{380\times0.4226}{0.0698}\approx\frac{160.588}{0.0698}\approx2299.26\)? Wait, no, wait, angle V is 151, angle T is 25, so angle U is 4? Wait, maybe I mixed up the sides. Wait, in triangle TUV, side u is opposite angle U, side t is opposite angle T. Wait, let's recheck. Angle T: 25°, angle V:151°, so angle U: 180 - 25 - 151 = 4°. Side u is opposite angle U (so length u is opposite angle U), side t is opposite angle T. So Law of Sines: \(\frac{t}{\sin T}=\frac{u}{\sin U}\). So \(t = \frac{u \times \sin T}{\sin U}\). So u = 380, sin T = sin25 ≈0.4226, sin U = sin4≈0.0698. So \(t = \frac{380\times0.4226}{0.0698}\). Wait, 380*0.4226 = 380*0.4 + 380*0.0226 = 152 + 8.588 = 160.588. Then 160.588 / 0.0698 ≈ 2299.26? That can't be right. Wait, maybe I mixed up the side labels. Wait, maybe side u is opposite angle V? Wait, no, standard notation: in triangle ABC, side a is opposite angle A, side b opposite angle B, side c opposite angle C. So in triangle TUV, side t is opposite angle T, side u opposite angle U, side v opposite angle V. So angle T: 25°, angle U:4°, angle V:151°. Side u (opposite angle U) is 380. So we need to find side t (opposite angle T). So Law of Sines: t/sin T = u/sin U. So t = (u * sin T)/sin U. So u=380, sin T=sin25≈0.4226, sin U=sin4≈0.0698. So t=(380*0.4226)/0.0698≈(160.588)/0.0698≈2299. But that seems too big. Wait, maybe the angle labels are wrong. Wait, maybe angle V is not 151, but maybe I misread. Wait, the problem says "m∠V=151° and m∠T=25°". So angle T:25, angle V:151, so angle U:4. Then side u is 380 (opposite angle U). So side t is opposite angle T. So the calculation is correct? Wait, but 380 is a small side (opposite 4°), and t is opposite 25°, which is larger, so t should be larger than 380. 2299 is larger, but maybe that's correct. Wait, but let's check with calculator. Sin25≈0.4226, sin4≈0.069756. So 380*0.4226=160.588. 160.588 / 0.069756≈2299.2. So to the nearest tenth, 2299.3? Wait, but maybe I made a mistake in angle identification. Wait, maybe the triangle is labeled differently. Maybe side u is opposite angle T? No, standard notation: vertex T, U, V. So angle at T: angle T, side opposite is UV (length t? No, wait, no: in triangle TUV, the sides: side opposite T is UV (length u? No, I'm confused. Let's re-express:

Let’s denote:

- Vertex T, angle at T: \( \angle T = 25^\circ \)

- Vertex U, angle at U: \( \angle U \)

- Vertex V, angle at V: \( \angle V = 151^\circ \)

Sides:

- Side opposite T: \( UV = t \)

- Side opposite U: \( TV = u = 380 \) inches

- Side opposite V: \( TU = v \)

Then, sum of angles: \( \angle T + \angle U + \angle V = 180^\circ \)

So \( 25^\circ + \angle U + 151^\circ = 180^\circ \)

\( \angle U = 180^\circ - 25^\circ - 151^\circ = 4^\circ \)

Law of Sines: \( \frac{UV}{\sin T} = \frac{TV}{\sin U} \)

So \( \frac{t}{\sin 25^\circ} = \frac{380}{\sin 4^\circ} \)

Thus, \( t = \frac{380 \times \sin 25^\circ}{\sin 4^\circ} \)

Calculating:

\( \sin 25^\circ \approx 0.422618 \)

\( \sin 4^\circ \approx 0.069756 \)

\( t = \frac{380 \times 0.422618}{0.069756} \)

\( 380 \times 0.422618 = 380 \times 0.4 + 380 \times 0.022618 = 152 + 8.59484 = 160.59484 \)

\( t = \frac{160.59484}{0.069756} \approx 2299.3 \)

Wait, but that seems extremely large. Maybe the problem has a typo, or I misread the angle. Wait, maybe angle V is 51° instead of 151°? Let's check. If angle V is 51°, then angle U is 180 -25 -51=104°, then t would be (380*sin25)/sin104 ≈ (380*0.4226)/0.9703≈(160.588)/0.9703≈165.5. That's more reasonable. Maybe a typo in the problem? But assuming the problem is correct as given (angle V=151°), then the calculation is as above. But maybe I mixed up the side. Wait, maybe side u is opposite angle V? Let's try that. If side u is opposite angle V (151°), then Law of Sines: \( \frac{t}{\sin T} = \frac{u}{\sin V} \). Then \( t = \frac{380 \times \sin 25^\circ}{\sin 151^\circ} \). Sin151°=sin(180-29)=sin29≈0.4848. Then t=(380*0.4226)/0.4848≈(160.588)/0.4848≈331.2. That's more reasonable. Ah! Here's the mistake: I misassigned the side. In standard notation, side u is opposite angle U, side v opposite angle V, side t opposite angle T. But maybe the problem uses a different notation, where side u is opposite angle V? Let's check the problem statement: "In \( \triangle TUV \), \( u = 380 \) inches, \( m\angle V = 151^\circ \), and \( m\angle T = 25^\circ \). Find the length of \( t \)". So in triangle TUV, the sides are labeled as follows: side t is opposite angle T, side u is opposite angle U, side v is opposite angle V. Wait, no, the lowercase letter is opposite the uppercase angle. So side t is opposite angle T, side u opposite angle U, side v opposite angle V. So angle V is 151°, so side v is opposite angle V. But the problem says u=380 inches. So side u is opposite angle U. So angle U is 4°, as before. But then the length is very large. Alternatively, maybe the problem has a typo, and angle V is 51° instead of 151°. Let's assume that's a typo (since 151° makes the other angle 4°, leading to a very long side). Let's recalculate with angle V=51°:

Angle U=180-25-51=104°

Law of Sines: \( \frac{t}{\sin 25^\circ} = \frac{380}{\sin 104^\circ} \)

Sin104°=sin(76)≈0.9703

t=(380*0.4226)/0.9703≈(160.588)/0.9703≈165.5. But since the problem says 151°, maybe I was wrong. Wait, let's check the original problem again. The user provided: "In \( \triangle TUV \), \( u = 380 \) inches, \( m\angle V = 151^\circ \), and \( m\angle T = 25^\circ \). Find the length of \( t \), to the nearest tenth of an inch."

So, side u is opposite angle U, angle V=151°, angle T=25°, so angle U=4°, side u=380. Then t is opposite angle T, so t=(u*sin T)/sin U=(380*sin25)/sin4. Let's compute with calculator:

Sin25≈0.4226182617

Sin4≈0.0697564737

380*0.4226182617=380*0.4226182617=160.5949394

160.5949394 / 0.0697564737≈2299.3

So that's the result. Maybe the triangle is very "flat" with a large angle, so the side opposite the small angle is short, and the side opposite the larger angle (25°) is much longer. So the answer is approximately 2299.3 inches? But that seems extremely long. Wait, maybe the unit is not inches? No, the problem says inches. Alternatively, maybe the angle is 15° instead of 151°? Let's check: angle V=15°, angle T=25°, angle U=140°, then t=(380*sin25)/sin140≈(380*0.4226)/0.6428≈(160.588)/0.6428≈249.8. Still, but the problem says 151°. So perhaps the problem is correct, and the answer is approximately 2299.3. But that seems odd. Wait, maybe I made a mistake in angle sum. 25 + 151 = 176, 180 - 176 = 4. Correct. So angle U is 4°. So the calculation is correct. So t≈2299.3 inches.

Answer:

\boxed{2299.3} (Wait, but let's check with calculator again. 380sin(25°)/sin(4°). Let's use more precise values. Sin(25°)=0.42261826174, sin(4°)=0.06975647374. 3800.42261826174=160.594939461. 160.594939461 / 0.06975647374=2299.336... So to the nearest tenth, 2299.3. So the answer is 2299.3.